An adaptive moving mesh method for thin film flow equations with surface tension

We present an adaptive moving mesh method for the numerical solution of thin liquid film spreading flows with surface tension. We follow the r-adaptive moving mesh technique which utilises a mesh density function and moving mesh partial differential equations (MMPDEs) to adapt and move the mesh coupled to the PDE(s) describing the thin film flow problem. Numerical experiments are performed on two one dimensional thin film flow equations to test the accuracy and efficiency of the method. This technique accurately resolves the multiple one-dimensional structures observed in these test problems. Moreover, it reduces the computational effort in comparison to the numerical solution using the finite difference scheme on a fixed uniform mesh. We are the first to implement r-adaptive schemes to higher order parabolic PDEs.This method accurately resolves the solution and reduces the computational effort.We introduce a new mesh density function that resolves multiple solution structures.This mesh density function is also adapted to include multiple solution components.